Are we getting interactions wrong?
The role of link functions
in psychological research

Laura Sità, Margherita Calderan, Tommaso Feraco,
Filippo Gambarota, Enrico Toffalini

Our dataset

  • 1,000 subjects
    • 500 typically developing children (group = 0)
    • 500 children with dyslexia (group = 1)
  • 50 trials per participant
  • Independent variable 1: age (in years)
  • Independent variable 2: group
  • Dependent variable: accuracy in a TRUE/FALSE task

Our dataset

Building the model

Key choices:

  • family
    specifies the response distribution and its valid range
    (e.g., unbounded, \([0,1]\), counts)

  • link function
    maps the linear predictor \(\beta_0 + \beta_1 \cdot age + \beta_2 \cdot group\)
    onto the scale of the response variable \(Y\)

Linear model

family=gaussian(link="identity")

Predictive check

New predicted values fall outside the valid range for accuracy [0,1]

family=gaussian(link="identity")

A positive interaction emerges

fit = glm(accuracy ~ age*group, family=gaussian(link="identity"), data=d)
summary(fit)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.492140   0.015673   31.40   <2e-16 ***
age          0.053481   0.001919   27.87   <2e-16 ***
group1      -0.377555   0.021786  -17.33   <2e-16 ***
age:group1   0.037318   0.002698   13.83   <2e-16 ***

Logistic regression model

family=binomial(link="logit")

family=binomial(link="logit")

Predictive check

New predicted values fall within the valid range for accuracy [0,1]

family=binomial(link="logit")

A negative interaction emerges

fit = glm(accuracy ~ age*group, data=d, family=binomial(link="logit"), weights= rep(k, nrow(d)))
summary(fit)

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -4.16402    0.19030 -21.881  < 2e-16 ***
age          0.87240    0.02599  33.573  < 2e-16 ***
group1       0.20253    0.23045   0.879    0.379    
age:group1  -0.14010    0.03139  -4.463 8.07e-06 ***

The appropriate model

The dataset was actually simulated

Code 
set.seed(123)

k = 50
N = 1000
group = rbinom(N,1,.5)
age = runif(N,6,10)
eta = -6+1*age-1*group  # linear predictor
probs = mafc.logit(.m = 2)$linkinv(eta)
accuracy = rbinom(n = N, size = k, prob = probs) / k

No interaction was simulated

Both models are detecting an interaction that does not exist

family=binomial(link=mafc.logit(.m=2))

To account for the 50% chance level in a TRUE/FALSE task:

2 alternatives forced-choice logit link

family=binomial(link=mafc.logit(.m=2))

* Logit(accuracy) with 0.5 as lower bound

family=binomial(link=mafc.logit(.m=2))

No interaction emerges, in line with how the data were generated

fit = glm(accuracy ~ age*group, data=d, family=binomial(link=mafc.logit(.m=2)), weights= rep(k, nrow(d)))
summary(fit)

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -5.773562   0.225527 -25.600   <2e-16 ***
age          0.975904   0.030091  32.432   <2e-16 ***
group1      -1.015102   0.299464  -3.390   0.0007 ***
age:group1  -0.006119   0.039296  -0.156   0.8763

Why interactions

link="identity"

Equal intervals on X correspond to equal intervals on Y

In our example the linear predictor is \(\beta_0 + \beta_1 \cdot age + \beta_2 \cdot group\)

link="logit"

Equal intervals on X correspond to equal ratios on Y when on the logit scale

link=mafc.logit(2)

Logit/probit on [chance level, 1] instead of [0,1]

Conclusions

Building a model means approximating the data-generating process (never observed directly in real data)

Key choices:

Family

Predicted values remain within the outcome’s valid range

Link function

Wrong links can create spurious interactions

Our systematic review of psychological research

How often

  • inappropriate link functions are used when testing interactions?

  • do they lead to significant results?

Materials & Contact

Data simulation, code and presentation are available on GitHub: sitalaura/link-functions

Questions and feedbacks: laura.sita@studenti.unipd.it

Bibliography

Domingue, B. W., Kanopka, K., Trejo, S., Rhemtulla, M., & Tucker-Drob, E. M. (2024). Ubiquitous bias and false discovery due to model misspecification in analysis of statistical interactions: The role of the outcome’s distribution and metric properties. Psychological methods, 29(6), 1164.

Hardwicke, T. E., Thibault, R. T., Clarke, B., Moodie, N., Crüwell, S., Schiavone, S. R., Handcock, S. A., Nghiem, K. A., Mody, F., Eerola, T., et al. (2024). Prevalence of transparent research practices in psychology: A cross-sectional study of empirical articles published in 2022. Advances in Methods and Practices in Psychological Science, 7 (4), 25152459241283477.

Liddell, T. M., & Kruschke, J. K. (2018). Analyzing ordinal data with metric models: What could possibly go wrong?. Journal of Experimental Social Psychology, 79, 328-348.

Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological bulletin, 105(1), 156.

Supplementary materials

Logit(accuracy) with link=logit

Logit(accuracy) with link=mafc.logit(.m=2)

* Logit(accuracy) with 0.5 as lower bound